Gelfond's constant

The constant appears in relation to the volumes of hyperspheres: () and surface areas ({{tmath|S_{n-1} }}) of -balls of radius . The volume of an n-sphere with radius is given by:V_n(R) = \frac{\pi^\frac{n}{2}R^n}{\Gamma\left(\frac{n}{2} + 1\right)},where is the gamma function. Considering only unit spheres () yields:V_n(1) = \frac{\pi^\frac{n}{2}}{\Gamma\left(\frac{n}{2} + 1\right)}, Any even-dimensional 2n-sphere now gives:V_{2n}(1) = \frac{\pi^n}{\Gamma(n+1)} = \frac{\pi^n}{n!}summing up all even-dimensional unit sphere volumes and utilizing the series expansion of the exponential function gives:\sum_{n=0}^\infty V_{2n} (1) = \sum_{n=0}^\infty \frac{\pi^n}{n!} = \exp(\pi) = e^\pi.We also have: If one defines andk_{n+1} = \frac{1 - \sqrt{1 - k_n^2}}{1 + \sqrt{1 - k_n^2}}for , then the sequence(4/k_{n+1})^{2^{-n}}converges rapidly to . ==Similar or related constants==

Ramanujan's constant The number is known as Ramanujan's constant. Its decimal expansion is given by: : = ... which turns out to be very close to the integer : This is an application of Heegner numbers, where 163 is the Heegner number in question. This number was discovered in 1859 by the mathematician Charles Hermite. In a 1975 April Fool article in Scientific American magazine, "Mathematical Games" columnist Martin Gardner made the hoax claim that the number was in fact an integer, and that the Indian mathematical genius Srinivasa Ramanujan had predicted it—hence its name. Ramanujan's constant is also a transcendental number. The coincidental closeness, to within one trillionth of the number is explained by complex multiplication and the q-expansion of the j-invariant, specifically:j((1+\sqrt{-163})/2)=(-640\,320)^3and,(-640\,320)^3=-e^{\pi \sqrt{163}}+744+O\left(e^{-\pi \sqrt{163}}\right)where is the error term,{\displaystyle O\left(e^{-\pi {\sqrt {163}}}\right) = -196\,884/e^{\pi {\sqrt {163}}}\approx -196\,884/(640\,320^{3}+744)\approx -0.000\,000\,000\,000\,75}which explains why is 0.000 000 000 000 75 below . (For more detail on this proof, consult the article on Heegner numbers.) The number The number is also very close to an integer, its decimal expansion being given by: : = ... The explanation for this seemingly remarkable coincidence was given by A. Doman in September 2023, and is a result of a sum related to Jacobi theta functions as follows: \sum_{k=1}^{\infty}\left( 8\pi k^2 -2 \right) e^{-\pi k^2} = 1. The first term dominates since the sum of the terms for k\geq 2 total \sim 0.0003436. The sum can therefore be truncated to \left( 8\pi -2\right) e^{-\pi}\approx 1, where solving for e^{\pi} gives e^{\pi} \approx 8\pi -2. Rewriting the approximation for e^{\pi} and using the approximation for 7\pi \approx 22 gives e^{\pi} \approx \pi + 7\pi - 2 \approx \pi + 22-2 = \pi+20.Thus, rearranging terms gives e^{\pi} - \pi \approx 20. Ironically, the crude approximation for 7\pi yields an additional order of magnitude of precision. The number The decimal expansion of is given by: : \pi^{e} = ... It is not known whether or not this number is transcendental. Note that, by Gelfond–Schneider theorem, we can only infer definitively whether or not is transcendental if and are algebraic ( and are both considered complex numbers). In the case of , we are only able to prove this number transcendental due to properties of complex exponential forms and the above equivalency given to transform it into , allowing the application of Gelfond–Schneider theorem. has no such equivalence, and hence, as both and are transcendental, we cannot use the Gelfond–Schneider theorem to draw conclusions about the transcendence of . However the currently unproven Schanuel's conjecture would imply its transcendence. The number Using the principal value of the complex logarithmi^{i} = (e^{i\pi/2})^i = e^{-\pi/2} = (e^{\pi})^{-1/2}The decimal expansion of is given by: : i^{i} = ... Its transcendence follows directly from the transcendence of and directly from the Gelfond–Schneider theorem. == See also ==